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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPoly_NthCoeff".
21 (* $Id: CPoly_NthCoeff.v,v 1.6 2004/04/23 10:00:53 lcf Exp $ *)
23 include "algebra/CPolynomials.ma".
26 * Polynomials: Nth Coefficient
27 %\begin{convention}% Let [R] be a ring and write [RX] for the ring of
38 alias id "R" = "cic:/CoRN/algebra/CPoly_NthCoeff/NthCoeff_def/R.var".
43 Notation RX := (cpoly_cring R).
49 The [n]-th coefficient of a polynomial. The default value is
50 [Zero:CR] e.g. if the [n] is higher than the length. For the
51 polynomial $a_0 +a_1 X +a_2 X^2 + \cdots + a_n X^n$ #a0 +a1 X +a2 X^2
52 + ... + an X^n#, the [Zero]-th coefficient is $a_0$#a0#, the first
53 is $a_1$#a1# etcetera. *)
55 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff.con".
57 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_strext.con".
59 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_wd.con".
61 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_fun.con".
65 We would like to use [nth_coeff_fun n] all the time.
66 However, Coq's coercion mechanism doesn't support this properly:
68 [(nth_coeff_fun n p)] won't get parsed, and has to be written as
69 [((nth_coeff_fun n) p)] instead.
71 So, in the names of lemmas, we write [(nth_coeff n p)],
72 which always (e.g. in proofs) can be converted
73 to [((nth_coeff_fun n) p)].
77 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nonConst.con".
80 The following is probably NOT needed. These functions are
81 NOT extensional, that is, they are not CSetoid functions.
84 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_ok.con".
86 (* The in_coeff predicate*)
88 inline "cic:/CoRN/algebra/CPoly_NthCoeff/in_coeff.con".
91 The [cpoly_zero] case should be [c [=] Zero] in order to be extensional.
94 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_S.con".
101 Implicit Arguments nth_coeff [R].
105 Implicit Arguments nth_coeff_fun [R].
109 Hint Resolve nth_coeff_wd: algebra_c.
113 Section NthCoeff_props
116 (*#* ** Properties of [nth_coeff] *)
118 alias id "R" = "cic:/CoRN/algebra/CPoly_NthCoeff/NthCoeff_props/R.var".
123 Notation RX := (cpoly_cring R).
128 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_zero.con".
130 inline "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_lin.con".
132 inline "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_lin.con".
134 inline "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_c_.con".
136 inline "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_O_x_mult.con".
138 inline "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_x_mult.con".
140 inline "cic:/CoRN/algebra/CPoly_NthCoeff/coeff_Sm_mult_x_.con".
143 Hint Resolve nth_coeff_zero coeff_O_lin coeff_Sm_lin coeff_O_c_
144 coeff_O_x_mult coeff_Sm_x_mult coeff_Sm_mult_x_: algebra.
147 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_ap_zero_imp.con".
149 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_plus.con".
151 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_inv.con".
154 Hint Resolve nth_coeff_inv: algebra.
157 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_c_mult_p.con".
159 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_p_mult_c_.con".
162 Hint Resolve nth_coeff_c_mult_p nth_coeff_p_mult_c_ nth_coeff_plus: algebra.
165 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_complicated.con".
167 inline "cic:/CoRN/algebra/CPoly_NthCoeff/all_nth_coeff_eq_imp.con".
169 inline "cic:/CoRN/algebra/CPoly_NthCoeff/poly_at_zero.con".
171 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_inv'.con".
173 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_minus.con".
176 Hint Resolve nth_coeff_minus: algebra.
179 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_sum0.con".
181 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_sum.con".
183 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_nexp_eq.con".
185 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_nexp_neq.con".
187 inline "cic:/CoRN/algebra/CPoly_NthCoeff/nth_coeff_mult.con".
194 Hint Resolve nth_coeff_wd: algebra_c.
198 Hint Resolve nth_coeff_complicated poly_at_zero nth_coeff_inv: algebra.
202 Hint Resolve nth_coeff_inv' nth_coeff_c_mult_p nth_coeff_mult: algebra.
206 Hint Resolve nth_coeff_zero nth_coeff_plus nth_coeff_minus: algebra.
210 Hint Resolve nth_coeff_nexp_eq nth_coeff_nexp_neq: algebra.